Pressure-sensitive dissipation in elastomers and its ...authors.library.caltech.edu/3734/1/KNAjap04.pdf · High explosives function by releasing large amounts of chemical energy when

Pressure-sensitive dissipation in elastomers and its implicationsfor the detonation of plastic explosives

Wolfgang G. Knauss and Sairam SundaramGraduate Aeronautical Laboratories, Mail Code 105-50, California Institute of Technology, Pasadena,California 91125

(Received 11 June 2004; accepted 21 September 2004)

The role of binder deformation and the associated energy dissipation on the detonation sensitivity ofplastically bonded explosives is considered by accounting for dilatation-sensitive viscoelastic shearresponse. Following the observation that pressurization can prolong the relaxation and retardationtimes of a viscoelastic elastomer tremendously, the implications of this phenomenon are consideredfor a thin layer of a model elastomer, sheared between two blocks of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine under deformation rates typical in detonation scenarios. Theconsequences of concurrent pressurization on heat generation are examined using small deformationas well as finite deformation analyses. While a dilatation-insensitive viscoelastic behavior generatesnotable temperature increases, they are insufficient to cause ignition of the explosive. However,taking into account the increased dissipation associated with the pressure-induced changes in theintrinsic time scale and viscosity of the elastomer leads to temperature rises on the order of 1000 °C,which are consistent with “hot spots” held responsible for the initiation of detonation in the adjacentexplosive grains. ©2004 American Institute of Physics. [DOI: 10.1063/1.1818349]

I. INTRODUCTION AND MOTIVATION

High explosives function by releasing large amounts ofchemical energy when heated to sufficiently hightemperatures.1 A common class of high explosives referred toas plastically bonded explosives(PBX) consist of grains ofan energetic material(e.g., octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine or hexahydro-1,3,5-trinitro-1,3,5-triazineheld together by a thin polymer matrix.2 The decompositionof the energetic grains releases energy and gaseous products,and the expanding gaseous products accelerate the rate ofreaction and strengthen the detonation front. In the case ofshock wave loading, however, the high temperatures requiredfor the initiation of chemical reaction in these explosivescannot normally be attained by a homogeneous deformationwithin the explosive. It is generally understood that in suchcases the initiation of detonation occurs in localized, smallregions within the explosive, referred to ashot spots, wherethe local temperatures are sufficiently high to initiate self-sustaining chemical reactions.

Several mechanisms have been proposed to explain theorigin of these hot spots. These include adiabatic heating oftrapped gases in cavities, local viscous heating due to voidcollapse, frictional rubbing between adjacent explosivegrains, and fracture of and shear banding in the explosivecrystals.3 Experimental evidence exists for almost all of thesemechanisms,4 and in any given situation involving the deto-nation of an explosive by impact, one or more of thesemechanisms may be dominant at various times during theinitiation to detonation transition. However, one othermechanism that is particularly relevant to the class of PBXhas not been studied in any significant detail. These materialsare composed of explosive crystals 20–200mm in dimen-sion, held together in an elastomeric matrix(typically a fewmicrons thick layer around the explosive crystals). This rub-

bery matrix comprises only a small fraction of the explosive(3%–10% by volume) but by this very fact, as will be de-scribed presently, can have a significant effect on the initia-tion of detonation in the explosive.

While it is generally acknowledged that localized sheardeformation mechanisms are among the leading contendersfor the formation of hot spots, much of the research on thisfront has been directed towards shear banding in the explo-sive crystals5,6 or frictional heating of shear cracksurfaces.7–9 However, several researchers have also sug-gested that hot spots can be generated by mechanical defor-mation of the polymeric binder phase.10,11There also appearsto be some experimental evidence for this mechanism of hotspot formation:12–14 Swallowe and Field10,13 and Heavensand Field12 report photographic studies of the deformation ofexplosive granules as well as polymers between impactingglass anvils and conclude that certain polymers exhibit asensitizing action on explosives. In drop-weight tests thesepolymers show a rapid reduction in the load required to pro-duce catastrophic failure of the sample. Postimpact micro-scopic examinations of the polymers reveal that the cata-strophic failure of the samples is associated with shear inlocalized bands and led the authors to conclude that thisfailure mode is responsible for the polymers’ sensitizing ac-tion. In a later section we discuss the connection betweenthis observation and the results of our computations.

Elastomeric binders used in high explosives typicallydisplay rubbery behavior at atmospheric conditions andhence possess characteristically low shear moduli(relative tothe bulk modulus). For example, a thermoplastic polyure-thane (commercially sold as Estane) used as a binder inmany high explosives, has a rubbery modulus of<2 MPa at100% elongation and 4 MPa at 300% elongation(B. F. Goo-drich data). Under shock wave loading, the thin binder layer

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between grains is subjected to high pressure and shear loads.Because binder is present as very thin layers sandwichedbetween harder explosive crystals, it can also experience lo-cally large shear strain rates.

Current mechanical models of explosives which attemptto describe the heterogeneous microstructure of plastic ex-plosives typically in terms of the rubbery behavior of thebinder.15 The associated low modulus values permit thebinder—and hence the explosive—to attain only small shearstress levels, which are associated with correspondingly lowinelastic dissipation values. On the other hand, it is knownthat elastomers undergo tremendous stiffening under im-posed pressure as a result of a change in the intrinsic timescale(see Fig. 1).

The influence of volumetric change on the rate or timedependent response of polymers is well recognized, thoughusually in connection with its effect on the value of the glasstransition temperature.16–19 Connected, at least qualitatively,with the notion of variability in free volume which changesthe degrees of freedom for motion of chain segments, smallchanges in volume can contribute to relatively large changesin the creep or relaxation response of polymers. This effecthas been explored for structural polymers below the glasstransition by Knauss and Emri20 and Losi and Knauss.21 Thenearly universal phenomenon of time-temperature trade-offfor linearly viscoelastic behavior has been characterized verysuccessfully by Tobolsky and Eyring22 and by Williams,Landel, and Ferry23 through a free volume formulation forthe time-temperature shift factor[defined in the well knownWilliams, Landel, and Ferry(WLF) equation].

We refer to this pressure induced effect as “stiffening,”because identical mechanical deformation within the mate-rial responds with a(much) higher modulus. This result is

the consequence of eliciting molecular response at a reducedinternal or intrinsic time scale in that the pressure transposesthe material towards or into the glassy state. If this transpo-sition is such that at the external(experimental) time scalethe material responds with the relaxation times mostly in themiddle of the rubber-to-glass transition, then it will, com-mensurately, produce maximal dissipation in any deforma-tion or relaxation process. Thus, what we refer to aspressure-induced stiffening is automatically associated withpressure-induced increases in the viscosity.

With respect to rubbery materials, Tschoegl andco-workers24,25 conducted extensive studies on the influenceof volume changes via mechanical pressure on the shear re-laxation moduli of several elastomers and have reportedmodulus increases of up to three orders of magnitude underpressure increases as low as 0.5 GPa. For example, Hypalon40 (a lightly filled chlorosulfonated polyethylene) exhibits ashear modulus of<1 MPa at 25 °C under atmospheric pres-sure. Under a pressure of 0.5 GPa the shear modulus in-creases to 630 MPa.26 By comparison, shock loading typi-cally raises pressure to levels on the order of a fewgigapascals before the initiation of detonation. Under suchconditions, the stiffening or hardening of the rubbery binderwould lead to much higher shear stress levels and corre-spondingly higher levels of inelastic dissipation in the binderthan response under atmospheric pressure would indicate.This elevated and pressure-augmented dissipation and conse-quent heating of the binder could produce local hot spots toignite the adjacent explosive crystals. Furthermore, the in-creased levels of shear stress sustained by the binder alsocause increased shear stresses in the explosive crystals andare likely to cause fracture of the crystals and thus form orsupport other sources of hot spots. On the other hand, in-

FIG. 1. Effect of pressure on the relaxation modulus in shear of the elastomer Hypalon 40[1 bar=curve(1); 4.6 kbar=curve(18)]. (Reproduced withpermission of the publisher Elsevier from Fillers and Tschoegl, Ref. 24).

J. Appl. Phys., Vol. 96, No. 12, 15 December 2004 W. G. Knauss and S. Sundaram 7255


creasing times and temperatures evoke the softer character-istics of the binder(response to the longer relaxation times)and thus tend to lower the stress and dissipation levels. It isthus the objective of this study to examine these competingeffects and their implications for the initiation of detonationsin plastic-bonded explosives. It may be noted here that thedata of Tschoegl and co-workers24,25 are from static experi-ments whereas the deformation of the binder in plastic ex-plosives takes place dynamically. The dynamic response ofpolymers under impact loading conditions is a subject of pastand current study(see, for example, Gupta27); however, suchexperiments still do not provide a comprehensive descriptionof polymer response over the large pressure-temperaturespace that would really be needed for the kind of analysisdescribed in this paper. We, therefore, base our study on thegenerally accepted understanding in polymer mechanics thatthe relaxation response of the polymer over a large time scalefrom nanoseconds to hours derived from the time-temperature-pressure superposition behavior under staticloading conditions is valid under dynamic conditions as well.Over the past half century there has been no investigationnor data that have contradicted this concept. Should futureexamination of this question materialize, we expect that de-viations from the current point of view are quantitativelylimited but without a total breakdown of the concept.

The paper is structured as follows. We begin with a briefreview of the experimental results of Tschoegl andco-workers24,25 (henceforth referred to as Tschoegl, Fillers,and Moonan or TFM) demonstrating the pressure-inducedincrease in the shear relaxation moduli(stiffening) of elas-tomers. These experimental characterizations are then usedin an analysis of the shearing of a thin, elastomeric binderlayer sandwiched between hard, explosive grains while un-der pressure. The problem is first analyzed in a small defor-mation framework which allows a clearer understanding ofthe primary effects, and then in a finite deformation frame-work. The purpose of this dual approach is to determine thatthe fundamental system response is not so much the result ofdetailed mechanics modeling but that suitable pressure sen-sitivity in the viscoelastic binder provides the dominant ef-fect. For both analyses, results of the calculations are pre-sented with discussions of the relevance to the detonation ofplastic explosives.

II. EXPERIMENTAL RESULTS ON PRESSURESENSITIVE RELAXATION

To the best of our knowledge Fillers and Tschoegl24 andMoonan and Tschoegl25 (TFM) have obtained the only com-prehensive measurements of the shear relaxation modulus ofelastomeric materials as a function of both pressure and tem-perature, though others28–30 have explored related effects inpolymers below their glass transition temperatures(rigidpolymers). Through an extensive set of measurements TFMdemonstrated that polymeric materials stiffen considerablywith increasing pressure by “translating them into a shortertime domain” and that the time-dependent modulus curvesdisplay a time-pressure superposition behavior similar to,and, essentially, in addition to, the well known time-temperature WLF superposition behavior. Specifically, seg-

ments of the time-dependent relaxation moduli curves of apolymer at different pressure levels are shifted relative toeach other along a logarithmic time axis and hence can berepresented by a single master curve and a pressure depen-dent shift factoraPsPd which represents the shifting of themaster curve with respect to the pressureP.

The combined effects of pressure and temperature arethen represented through a master curve at a reference tem-perature and reference pressuresT0,P0d combined with apressure and temperature dependent shift functionaT,PsT,Pd.Such a material is referred to as a thermorheologically- andpiezorheologically simple material. Figure 1 shows a set of(isothermal) shear relaxation modulus measurements at25 °C and different pressures for Hypalon 40. Figure 2shows the corresponding master curve reduced toT0

=25 °C andP0=1 bar resulting from the shifting of isother-mal and isobaric curves in Fig. 1. The inset shows the cor-responding shift functionaT,P.

For those not routinely versed in thermoviscoelastic be-havior of polymers a comment is in order that addresses therelation between the material characterization effected quasi-statically in the Fillers/Tschoegl work and the dynamicalconditions associated with the high rate pressurization prob-lem typical for explosives. One first notes that, in principle,constitutive behavior is formulated independently of themode or speed of deformation: Inertial considerations do notenter into the description of constitutive relations. Like anyelastic material characterization, viscoelastic material de-scriptions apply equally to static and dynamic problems. Thedifference between the two sets of materials is that the latterare implicitly deformation rate sensitive regardless ofwhether material inertia plays a role or not, and the same isnot true for the former.

On the other hand, an approximation commonly used inengineering problems where volumetric deformations are en-countered, will be employed here. This approximation treatsthe bulk response as elastic so that no delayed volumechanges occur under rapid pressure applications. This is amild substitution for real material behavior because the vis-coelastic effect on volume deformation is orders of magni-tude smaller than the shear behavior.31–33

We write the linearly viscoelastic constitutive descrip-tion as

si j = 2E−`

t

m0st − jd]ei j

]jdj + di jE

−`

t

K0st − jd]ekk

]jdj, s1d

wherem0std andK0std are the time-dependent shear and bulkrelaxation moduli at some reference temperature and pres-suresT0,P0d.

If pressure and temperature change during the course ofthe deformation, the material relaxation times are affectedincrementally in time to produce a stiffening or acceleratedsoftening of the binder in addition to the normally intrinsictime-dependent relaxation. The effect of changing tempera-ture and pressure is included through the concept of anin-trinsic material timedefined as

7256 J. Appl. Phys., Vol. 96, No. 12, 15 December 2004 W. G. Knauss and S. Sundaram


t8 =E0

t dt

aT,P, s2d

where

aT,P = aT,PsT,Pd = aT,PfTstd,Pstdg s3d

is the temperature and pressure dependent shift function asdefined with respect to the reference conditions on tempera-ture and pressuresT0,P0d (e.g., see inset in Fig. 2). The(nonlinear) viscoelastic formulation is then written as

si j = 2E−`

t

m0st8 − j8d]ei j

]jdj + di jE

−`

t

K0st8 − j8d]ekk

]jdj.

s4d

The expression for the shift functionaT,P given by TFMis

ln aT,P = −c1hfT − usPdg − T0jc2 + fT − usPdg − T0

, s5d

where

usPd = c3 lnF 1 + c4P

1 + c4P0G − c5 lnF 1 + c6P

1 + c6P0G . s6d

The reader is referred to Refs. 24 and 25 for details on thederivation of Eqs.(5) and (6) as well as on obtaining thevalues of the parametersci.

A. Modification of the TFM shift function for higherpressures

The TFM expression for the temperature and pressuredependent shift functionaT,P [Eq. (5)] is a modification ofthe WLF (Ref. 23) expression for the shift function whichcan be based on free volume changes in connection withtemperature changes. The TFM expression includes the ef-fect of pressure on free volume through the compressibility,and is incorporated into the WLF expression as an apparentpressure-induced temperature drop denoted byusPd. Increas-ing the pressure is, thus, equivalent to decreasing the tem-perature. However, a feature of the free volume based WLFtheory is that for temperatures below the glass transition tem-peratureTg the free volume remains(nearly) frozen at itsvalue corresponding toTg. Correspondingly, the value of theshift function does not change much belowTg. The implica-tion for the TFM model would be that, forfT−usPdgøTg,the value of lnaT,P remains saturated at or near the value

ln aTg,P = −c1hfTg − usPdg − T0jc2 + fTg − usPdg − T0

. s7d

The modified TFM shift function is schematically illustratedin Fig. 3.

For the material whose master relaxation modulus curveis shown in Fig. 2, the glass transition temperature(at atmo-spheric pressureP0) is <−20 °C so that, at the referencetemperatureT0s=25 °Cd, the shift function might be ex-pected to saturate at a pressure corresponding tousPd=T0

−Tg=45 °C. The value of the pressure corresponding to thisvalue ofusPd is <0.4 GPa and it can be seen from Fig. 2 thatthe shift function indeed begins to deviate from the TFM

FIG. 2. Master relaxation curve and shift function for Hypalon 40(from Fillers and Tschoegl, Ref. 24). This is a single curve representation of the pressureeffects shown in Fig. 1 and also includes the effect of temperture. The inset figure shows the shift functionaT,P that, together with the master curve,summarizes the effect of temperature and pressure on the shear relaxation modulus of the elastomer.



expression near this pressure. This modification to the TFMexpression cannot be overlooked for the present purposesbecause the stress states of primary concern typically involvepressures higher than 0.4 GPa and use of the referenced,nonmodified TFM expression would be likely to cause unre-alistic pressure-induced shifting of the relaxation behavior.

In the following section, we consider the problem of athin layer of elastomeric binder being sheared under imposedpressure between two relatively hard(semi-infinite) grains ofexplosive. We examine this first in the context of small de-formations with the constitutive characterization given byEq. (4).

III. SMALL DEFORMATION ANALYSIS OF BINDERDEFORMATION

When a (granular) composite such as PBX deforms,highly inhomogeneous deformations occur at the scale of theexplosive grains. Where grain-to-grain contact is not thedominant form of force or stress transmission. Deformationgradients are highest between grains, especially under localshear deformations, since these involve the lowest stiffnesscharacteristics of the composite. Consider then a thin layer ofviscoelastic binder sandwiched between two relatively largegrains of explosive and undergoing shear deformation(Fig.4). Our objective is to evaluate the dynamic stress-deformation behavior of this analog binder material. Thisincludes the evaluation of the temperature increase in thebinder as a result of viscoelastic and pressure-augmented dis-sipation as well as the resulting temperatures in the adjacentexplosive grains. The binder generally forms a thin layerbetween explosive grains and is subject to stress-waves trav-eling through the explosive composite. The transit time ofthese waves(pressure jump) through the binder layer is typi-cally on the order of a few nanoseconds whereas the timescales leading to detonation are on the order of tens of mi-croseconds. Hence it is appropriate to study the homoge-neous shear deformation behavior of the binder. In the fol-lowing development, the nonlinear constitutive relation forthe polymeric binder expressed as an integral formulation inEq. (4) will be rederived in rate form to facilitate the numeri-

cal evaluation of the inelastic dissipation rate which deter-mines the temperature increase in the binder duringdeformation.

A. Rate equations and constitutive modeling

Let the time-dependent behavior of the viscoelasticbinder at the reference temperature and pressuresT0,P0d becharacterized by a shear relaxation modulus functionGstd inthe form of a Prony series

Gstd = oi=1

N

Gi exps− t/ti0d. s8d

Here ti0 are specific relaxation times at the reference

temperature and pressuresT0,P0d andGi are the correspond-ing moduli. The material may have, in fact, a very largenumber of individual relaxation mechanisms and corre-sponding relaxation times, but for realistic computationalpurposes only a limited number are necessary to adequatelymodel the overall relaxation behavior over the entire time-scale of the relaxation process. Furthermore, for a ther-morheologically and piezo-rheologically simple material therelaxation times at some other temperature and pressuresT,Pd are given by

tisT,Pd = ti0aT,P, s9d

where aT,P=aT,PsT,Pd is the shift function of Eq.(3) dis-cussed in the preceding section. The representation of Eq.(9)corresponds to a shifting of the lnfGstdg−lnftg curve(Fig. 2)by lnsaTPd along the lnftg axis. We again note here that, ingeneral, higher temperatures tend to accelerate relaxationwhereas increased pressures have the opposite effect.

For representation purposes, consider a Maxwell-Wiechert model as shown in Fig. 5 consisting ofN parallelMaxwell elements. LetGi represent the individual springmoduli andhi

0 the individual dashpot viscosities. If one de-fines

FIG. 3. Proposed modification to the TFM shift function[Eq. (5)] to accountfor saturation of pressure-stiffening effects at higher pressures(as well astempertures belowTg).

FIG. 4. Configuration for small deformation analysis of binder shearing. Athin binder layer is sandwiched and sheared between two layers of explosivegrains under pressure P and shear strain ratee0.



ti0 = hi

0/Gi , s10d

then the overall relaxation modulus in shear of the model isgiven by Eq.(8) and hence we use this Maxwell-Wiechertmodel as the mechanical spring-dashpot analog for thebinder to calculate the rate of viscoelastic energy dissipationduring the binder deformation.

Since the individual spring-dashpot elements are in par-allel, each element undergoes identical deformation. The in-dividual dashpot viscosities, however, change during the de-formation according to

histd = hi0aT,Pstd, s11d

which is equivalent to Eq.(9).In the following analysis, shear stresses are denoted bys

instead oft to avoid confusion with the relaxation timesti.Correspondingly, shear strains are represented bye. Let theoverall rate of straining of the assembly at any instant beestd. Then, for each spring-dashpot elementi we write

estd = eiGstd + ei

hstd, s12d

whereeiG and ei

h represent the rate of straining of the springand dashpot element, respectively.

If si represents the stress in this element, then

eiGstd =

sistdGi

s13d

and

eihstd =

sistdhi

0aT,Pstd=

sistdGiti

0aT,Pstd. s14d

Hence, combining Eqs.(12), (13), and(14) results in

estd =sistdGi

+sistd

Giti0aT,Pstd

, s15d

which may be rewritten as

sistd = Giestd −sistd

ti0aT,Pstd

. s16d

Equation (16) can be integrated numerically to givesistd.The rate of viscoelastic dissipation in theith element is then

Wivstd = sistdei

hstd =fsistdg2

Giti0aT,Pstd

s17d

and the total stress and dissipation rate in the material arefound, respectively, as the sums

sstd = oi=0

N

sistd s18d

and

Wvstd = oi=0

N

Wivstd = o

i=0

Nfsistdg2

Giti0aT,Pstd

. s19d

If we assume, for the moment, that there is no conduc-tion of heat to the adjacent explosive grains, the rate of tem-perature rise of the homogeneously deforming binder isgiven by

Tstd = bWvstdsrCdb

, s20d

wherer andC are the density and specific heat capacity ofthe binder, respectively, both of which are taken to be con-stant with respect to time, temperature, and pressure for pur-poses of this calculation andb is the fraction of the inelasticdissipation contributing to heating of the binder. For pur-poses of our analysis we assume that all inelastic dissipationis converted to heating of the bindersb=1d. The temperaturehistory of the binder is then represented by

Tstd = T0 +E0

t WvsjdrC

dj. s21d

A Prony series representation of the form of Eq.(8) isevaluated for the master relaxation curveGstd in Fig. 2 bychoosing 13 relaxation times spaced approximately one de-cade apart. The resulting thirteen individualGis andti

0s arelisted in Table I. The densityr for the binder, under atmo-spheric conditions, is 1150 kg/m3 and the specific heat ca-

FIG. 5. Maxwell-Wiechert model representation of the mechanical behaviorof the binder corresponding to the Prony series representation of Eq.(8).

TABLE I. Values of Prony series parametersti0 and Gi used to fit the

experimental shear relaxation modulus data for Hypalon 40(see Ref. 24 andFig. 2).

ti0

(s)Gi

(Pa)

0.100310−8 0.1103109

0.838310−8 0.8033108

0.702310−7 0.7123108

0.588310−6 0.2133109

0.492310−5 0.2043109

0.412310−4 0.6583108

0.346310−3 0.8333107

0.289310−2 0.2173107

0.242310−1 0.3953106

0.203310+0 0.4003106

0.170310+1 0.1003106

0.143310+2 0.4003106

0.119310+5 0.9003106



pacity C is 300 J/skg Kd and taken to be independent oftemperature and pressure for reasons of simplicity.

B. Temperature and energy considerations

As the binder undergoes deformation, the viscous dissi-pation causes the temperature of the binder to increase. Ifone assumes that no heat is lost to the adjacent explosivegrains, the temperature increase of the binder may be evalu-ated from the evolution equation

Tb =Wv

srCdb, s22d

whereWv is the total dissipation rate andsrCdb is the heatcapacity per unit volume of the binder. However, to deter-mine the feasibility of forming hot spots in the region aroundthe deforming binder, it is necessary to consider the heatconduction into adjacent explosive grains. Thus, to properlyestimate the temperature distribution in the explosive grainsadjacent to the binder, it is, in principle, necessary to alsosolve the associated heat conduction equation in the grainssubject to appropriate boundary conditions at the binder-explosive interface. Such an approach would also necessitatea spatial solution of the temperature field within the binderthickness. However, since we have resorted to a consider-ation of homogeneous deformation of the binder as a way ofreducing the complexity involved in a full solution withoutsacrificing much by way of physical insight and quantitativeunderstanding, we present, in the same spirit, a simplified butsufficiently accurate treatment of the conduction issue.

It can be estimated from basic heat diffusion theory thatat any timet about 90% of the heat content in the explosivegrain will be contained within a distance ofÎ4aet from theinterface with the binder, whereae is the thermal diffusivityof the explosive grain. For the time span of 10ms for whichcomputational results are evaluated later on, this distance is<3 mm. Since the binder layer thicknesshb is on the orderof 20 mm, it is reasonable to assume that no significantamount of conduction will have occurred far into the explo-sive grains to warrant a full heat diffusion analysis. The tem-perature of the explosive grain just adjacent to the binderwill thus be very close to the binder temperature evaluatedby using the adiabatic rate equation Eq.(22). However, weaccount for heat conduction approximately by considering auniformly heated zone of lengthhestd=0.5Î4aet into the ex-plosive grains(see Fig. 6). Thus, the temperature history ofthe binder and explosive just adjacent to the binder may becalculated from

T =hbW

v

hbsrCdb + 2hestdsrCde. s23d

We next account for the chemical energetics of the ex-plosive grains. Iff represents the mass fraction of reactionproducts then the energy release per unit volume due tochemical reaction is given by

We = freDH, s24d

whereDH represents the heat of detonation, and

f = s1 − fdn exp −DG

kTs25d

represents the progress of the reaction in terms of single-stepArrhenius kinetics. In the latter expression,DG signifies anactivation energy for the reaction andn is a pre-exponentialfactor generally identified with a characteristic vibrationalfrequency for the atoms in the explosive grains. Since ourobjective is not a detailed calculation of detonation reactionzones, but rather a study of characteristic features of theresponse of the energetic solid, we consider this one-stepenergetics representation sufficient. The temperature historyof the binder and the explosive grains just adjacent to thebinder is now determined by

T =hbW

v + 2hestdWe

hbsrCdb + 2hestdsrCde. s26d

C. Computational results and discussions

The evolution equations of the previous section are con-verted to a finite-difference form for numerical implementa-tion. A time step of 1 ns is used throughout the analysis anda total deformation time of 10ms is considered since, typi-cally, detonations in explosives initiate at time scales on theorder of microseconds. The relevant parameter values aretaken as follows:34,8 hb=20 mm, DH=5.53106 J/kg, DG=2.0 eV,n=1014 s−1, rb=1150 kg/m3, re=1900 kg/m3, Cb

=300 J/skg Kd, Ce=970 J/skg Kd, and ae=0.28310−6 m2/s.

Figures 7(a) and 7(b) show the results for a constantstrain rate deformation of 0.53106 s−1 under an imposedpressure of 0.3 GPa. These plots demonstrate the effect ofaccounting for the stiffening and/or increased dissipation ofthe binder under the imposed pressure. Each plot shows theshear stress and temperature history at the interface betweenthe explosive and the binder during the course of the defor-mation.

When pressure-augmented dissipation is not accountedfor [Fig. 7(a)], the temperature near the interface increasesuniformly with time but does not rise to values sufficient for

FIG. 6. Approximate thermal analysis for heat conduction from the deform-ing binder layer into the adjacent explosive grains. The dimensionhestdrepresents the approximate thickness of the layer in the explosive grainswhere most of the conducted heat resides at any timet.



initiating any significant chemical decomposition of the ex-plosive in the time of 10ms. On the other hand, when thepressure-induced stiffening or dissipation increase is takeninto account[Fig. 7(b)], a much higher shear stress is ini-tially sustained by the binder because of the pressure-induced stiffening but initially with only very little dissipa-tion and a correspondingly small temperature rise. However,around 1.5ms, the binder softens rapidly under the influenceof the rising temperature, leading to a fast and significantfurther heating that arises from the inelastic dissipation ofmost of the energy stored “elastically” in the binder untilthen. This temperature increase to about 900 K is sufficientto set off the chemical energetics of the explosive so as tolead to a “thermal explosion” at around 6ms. As the ener-getic material is exhausted by decomposition, the reactionprogresses to completion, and the thermal explosion sub-sides.

It should be noted here that this drop in the shear carry-ing capacity of the binder at<1.5 ms is possibly relevant tothe experimental observations reported by Field andco-workers10,12,13 on the catastrophic load drop for PBX inconnection with drop-weight tests. As discussed in the intro-duction, these researchers concluded that this “catastrophic”failure of polymers along localized bands is responsible fortheir “sensitizing action” in explosives, without furtherspecifying the nature of this “sensitization.”

Figures 8(a), 8(b), 9(a), and 9(b) represent excerpts of aparameter study with respect to the relevant loading param-eters, pressure, and shear deformation rate. Having estab-

FIG. 8. Shear stress(solid curve) and temperature(broken curve) historiesat the binder/explosive grain interface for the small deformation analysiswith pressure sensitivity and withP=0.3 GPa. Effect of strain rate.(a) e0

=0.13106 s−1. (b) e0=0.73106 s−1.

FIG. 9. Shear stress(solid curve) and temperature(broken curve) historiesat the binder/explosive grain interface for the small deformation analysiswith pressure sensitivity and withe0=0.53106 s−1. Effect of pressure(a)P=0.2 GPa.(b) P=0.1 GPa.

FIG. 7. Shear stress(solid curve) and temperature(broken curve) historiesat the binder/explosive grain interface withP=0.3 GPa ande0=0.53106 s−1 for the small deformation analysis.(a) No pressure sensitivity ofviscoelastic response.(b) Pressure sensitivity included. Including pressuresensitivity significantly alters the shear response and causes a rapid tempera-ture rise in the binder upon catastrophic softening of the binder.



lished the necessity to properly account for pressure-augmented dissipation in the rubbery binder, thesecomputations all include pressure sensitivity. Figures 8(a)and 8(b) show, respectively, the effect of a lower and highershear deformation rate relative to that represented in Fig.7(b) with the pressure held at 0.3 GPa. With the lower de-formation rate[Fig. 8(a)] insufficient elastic energy has ac-cumulated when the shear stress drops to cause the tempera-ture to rise as high as in Fig. 7(b). The temperature rise issignificant s<125 °Cd but is insufficient to activate thechemical energetics. At a higher deformation rate[Fig. 8(b)],more elastic energy has accumulated at the time the bindersoftens rapidly and hence the temperature rise is higher andthe energetics are sped up as well. Figures 9(a) and 9(b)show the effect of pressure with the same shear deformationrate as Fig. 7(b). As the pressure reduces, the stiffening anddissipation effects reduce considerably, also. Higher pres-sures are not considered up to this point in the discussionbecause the constitutive formulation needs modification toaccount for other inelastic deformation mechanisms at thehigher stress levels caused by the greater stiffening at higherpressures. This point is discussed subsequently in moredetail.

IV. FINITE DEFORMATION ANALYSIS OF BINDERDEFORMATION

The small deformation analysis presented in the previoussections allows an understanding of the main characteristicsof the shear response of the viscoelastic binder under com-bined pressure and shear loading. In view of the large defor-mations experienced by the binder under these loading con-ditions, it is appropriate to also inquire whether finitedeformation effects change the results significantly. How-ever, today there is no generally accepted or experimentallyverified constitutive description for large deformation vis-coelastic behavior, so that the following developmentsshould be viewed as tentative in precision though basicallycorrect in terms of global behavior. The following analysisclosely follows that of Tonget al.35 to which publication thereader is referred for more detail.

A. Deformation kinematics

We consider again the case of homogeneous shear defor-mation of a thin binder layer between two hard explosivegrains. Letlstd andkstd represent the stretch and shear of thebinder at any time during the deformation(Fig. 10) so thatthe deformation gradient tensor can be written as

F = 1 lstd 0 0

− kstd 1 0

0 0 12 . s27d

From the deformation gradient tensor the spatial velocitygradient is obtained as

L = FF−1 = D + W , s28d

whereD=sL +L Td /2 is the rate of deformation tensor, andW =sL −L Td /2 is the spin rate tensor.

The first Piola-Kirchhoff stress tensorT is related to theCauchy stress tensors by

s =1

JFTT, s29d

where J=detF. From the symmetry of the Cauchy stress,one obtains

FTT, = TFT, s30d

which identifies the first Piola-Kirchhoff stress tensor to beof the form

T = 1T11 kT11 + lT21 0

T21 T22 0

0 0 T332 . s31d

B. Constitutive relations and rate equations

We make the typical large-definition assumption that thedeformation gradient tensor allows a multiplicative decom-position into elastic and inelastic(viscous) components ofthe form

F = FeFv. s32d

The above representation is refined by assuming that in aviscoelastic material there exist several ‘independent’ butsimilar and concurrently active relaxation mechanisms, eachof which is governed by the overall deformation gradientF.Thus we write36

F = Fi = FieFi

v, s33d

where Fie and Fi

v represent the elastic and viscous compo-nents of individual mechanisms identified by the subscripti.Unless indicated, there is no sum over the subscripti. Thephysical motivation for this representation follows from the

FIG. 10. Configuration for finite deformation analysis of binder shearing.kand l represent the two kinematic measures of deformation of the binder,respectively, the shear and the stretch.



Maxwell-Wiechert model used in the small deformationanalysis(see Fig. 5).

The multiplicative decomposition ofF, together with Eq.(28) then leads to

D = Di = Die + Di

v, s34d

which can be rewritten as

Die = D − Di

v. s35d

Thus if the viscous componentDiv of the rate of deformation

tensorDi is related to the stress tensor using an appropriateinelastic flow law, and the elastic componentDi

e is related tothe stress rates, one can obtain the evolutionary rate equa-tions for the components of the stress tensor.

Rather than assuming a general Neo-Hookean hyperelas-tic response, we employ a linear response for the elastic be-havior in this analysis. As discussed later, this assumptiondoes not significantly affect the features of the behavior ofthe binder under the loading conditions considered; at thesame time it minimizes the complexity of the formulationand focuses attention on the more relevant relaxation behav-ior.

The viscous flow rule for each relaxation mechanismi istaken to be in the form of the associated flow law

Div = gi

v Si

2sief f , s36d

where Si =si −1/3stracesiidI is the deviatoric stress tensor,si

ef f=Î1/2SiklSi

kl (sum overk and l) is the effective stress,and gi

v is the viscous or plastic strain rate function.The specific form of the plastic strain rate functiongi

v isobtained by again referring to the Maxwell-Wiechert modelfrom which we may write

Div =

Si

Gitistd, s37d

which is the large deformation version of Eq.(14).Comparing this relation to Eq.(36) above renders

giv =

2sief f

Gitistd, s38d

where, as before,tistd=ti0aT,Pstd are the temperature and

pressure-dependent relaxation times.The resulting rate equations for the componentsTi

ab ofthe first Piola-Kirchhoff stress tensorT i for each mechanismi can then be written as

Tiab = Ai

abl + Biabk − Ri

abgiv, s39d

where

Aiab = Ai

absT i,l,kd, s40ad

Biab = Bi

absT i,l,kd, s40bd

Riab = Ri

absT i,l,kd s40cd

are lengthy algebraic expressions listed in the appendix. Itcan be seen that the first two terms on the right-hand side ofthe rate expression Eq.(39) represent the finite-deformation

elastic contributions due to the overall deformation rateslandk, and the third term represents the stress relaxation dueto the viscous flowgi

v. The reader is referred to Ref. 35 formore detailed discussions on this finite deformation analysis.

The overall stress in the binder is calculated as

T = oi=1

N

T i , s41d

whereN is the total number of relaxation mechanisms con-sidered(see Table I), and the overall viscous dissipation rateis given by

Wv = oi=1

N

Wiv = o

i=1

N

Si:Div = o

i=1

N

sief fgi

v. s42d

C. Loading and boundary conditions

In the computational results presented below, the behav-ior of the binder is examined for the case of wave loading by

prescribing the initial deformation ratesl0 andk0 and stress-velocity boundary conditions which result in the followingrate equations forl andk:

l = l0 −2

hbsrc1deT11, s43ad

k = k0 −2

hbsrc2deT21, s43bd

with hb denoting the thickness of the binder layer andsrc1de

and src2de the longitudinal and shear impedances of the ex-plosive grains adjacent to the binder. The initial deformation

rates l0 and k0 are related to the particle velocity of theincoming wave in the explosive grains and the thickness ofthe binder layer. In the interest of brevity, the derivation ofthese equations is not discussed but the reader is referred tothe discussions of wave loading in Ref. 35 for further under-standing. Equations(43) along with Eq.(39) provide a com-plete set of rate expressions for updating the stress compo-nents. The thermal and energetics analysis is essentially thesame as that used in the small deformation analysis.

D. Computational results and discussion

The rate equations for stress, deformation, and tempera-ture are again solved numerically using the finite differencetechnique. A time step of 1 ns is again used throughout andcomputations extend over 10ms of deformation. The initialvalues for the deformation rates are taken to bek0=1.0

3106 s−1 and l0=−0.33106 s−1. Other values are the sameas those used in the small deformation analysis.

Figures 11(a) and 11(b) show the results of the finitedeformation computations using these parameters. Each plotshows the shear stressT21 (solid curve) and temperature his-tory at the interface between the explosive and the binder(broken curve) during the course of the deformation. Thebehavior is very similar to that observed for the small defor-mation analysis[Figs. 7(a) and 7(b)] and again demonstratethe dominating effect of the stiffening of and dissipation in



the binder under the(evolving) pressure. For completenessof presentation we show in Fig. 12 the evolution history ofthe pressureP, shift functionaT,P, and the shear deformationrate k. The pressure initially rises rapidly due to the com-pressive wave and induces a corresponding stiffening of theshear modulus(experienced as an increase in the value of theshift function aT,P). However, in the brief time it takes thepressure to build up(by wave reverberation within the binderlayer), the binder undergoes intrinsic time-dependent soften-ing, and the accompanying viscous heating triggers a rever-sal in the pressure-induced stiffening of the binder(decreas-

ing value ofaT,P). As the temperature builds up, this thermalsoftening is accelerated until at about 2.5ms the binder un-dergoes catastrophic softening and a corresponding rapid re-duction in the shear stress.

V. FURTHER DISCUSSION

The main objective of this work was to showcase theneed for a proper accounting of the pressure sensitivity of theconstitutive response of polymers apart from the relativelycommon considerations addressing the temperature and theintrinsically time-dependent response of binder materials.The need for a properly combined consideration of the time,temperature, and pressure-dependent response is particularlyrelevant to rubbery materials under shock loading conditions.During the short time-scales relevant to shock loading con-ditions, the normally “rubbery” material has, initially, accessonly to its (orders of magnitude stiffer) glassy response.However, as the temperature increases under the accompany-ing inelastic deformation, the material is able to access, pro-gressively, the entire spectrum of its time-dependent re-sponse due to the strong, temperature-driven acceleration ofthe relaxation processes. Including the correspondingpressure-driven stiffening of the material, along with itspressure-governed viscosity, produces then a competition be-tween the time, temperature, and pressure-dependent effectsso as to generate responses that are significantly differentfrom those predicted without such detailed considerations. Inthis work we have demonstrated these various competingeffects by analyzing the response of the thin, rubbery binderin plastic explosives under loading conditions that occur dur-ing the initiation and propagation of detonation.

At this point it is relevant to summarize the major fea-tures of the analysis, under review of the assumptions andbroad conclusions and to point to caveats derived from thepresently restricted understanding of polymer behavior.

(1) Although temperature, and pressure-related nonlin-earity is accounted for through a modification in the timescale, the reference master curve for the relaxation modulusis based on small-stress, small-strain, linearly viscoelasticbehavior. Nonlinearity arising from larger stresses and strainsis yet poorly characterized or understood in general. How-ever, the major relevant feature is the decrease of the shearrelaxation modulus by threeorders of magnitudeover sev-eral decades in time and hence modeling the relativelysmaller nonlinear effects is not expected to change theprominent effects of the response in the analysis.

(2) To focus attention on the relaxation effects and avoidunnecessary complexity, the rubber elasticity has been mod-eled as linear rather than derived through a hyperelastic po-tential. This is justifiable because in the early stages of thedeformation when the main effects(rapid fall in shear stressand corresponding viscous heating of the binder) are mani-fest, the short-time relaxation moduli are closer to the glassyvalues and hence orders of magnitude larger than any non-linear modulus effects; at longer times when these effectsmay be significant, the stress levels are not high enough torender the nonlinear modulus effects significant in the overall

FIG. 11. Shear stress(solid curve) and temperature(broken curve) histories

at the binder/explosive grain interface withk0=1.03106 s−1 and l0=−0.33106 s−1 for the finite deformation analysis.(a) No pressure sensitivity ofviscoelastic response.(b) Pressure sensitivity included. The response issimilar to that observed with the small deformation analysis(Fig. 7).

FIG. 12. Evolution ofP, ln aT,P, and k for the finite deformation analysiscorresponding to the results shown in Fig. 11(b). The pressureP saturates at<0.45 GPa and the rate of shear deformationk decreases from the initialloading value as the binder offers resistance and subsequently increasesagain as the iner softens. The shift functionaT,P shows the initial(stiffening)due to the pressure followed by the decrease(softening) as a result of theincreasing temperature.



context of the model. These nonlinear modulus effects arepotentially less significant than the unknown nonlinear ef-fects discussed in(1) above.

(3) Again, in an effort to keep attention focused on re-laxation and dissipation effects and to minimize analyticalcomplexity, a simplified treatment of heat conduction hasbeen adopted. This approach is permissible, firstly becausethe conduction distances into the explosive grains during thetime scale of the detonation process are much smaller thanthe binder thickness, and secondly because the sudden tem-perature increase due to the corresponding rapid drop inshear stress occurs over a very short period of time andhence can be considered to be(very nearly) adiabatic. Theonly possibly significant effect of the simplified analysis willbe in the time to “thermal explosion” after this sudden tem-perature increase, but in this regard the analysis providesconservative(longer time) estimates for the time to explo-sion because peak temperatures near the binder-grain inter-face will be slightly higher than the average temperatureused in the analysis.

VI. CONCLUSION

The present investigations indicate that if pressure-induced stiffening or dissipation augmentation of the vis-coelastic binder is accounted for along with a proper descrip-tion of the time and temperature-dependent response, themechanism of shear-induced inelastic dissipation in the ex-plosive binder can cause the explosive to be heated locally totemperatures that are sufficient to lead to detonation within afew microseconds. This phenomenon may thus be viewed asa plausible mechanism of hot spot formation. At low shockstress levels, only a few binder layers may be oriented favor-ably to make this mechanism feasible and other hot spotmechanisms may be more dominant, but as detonationprogresses and the shock strengthens, this mechanism maybe activated at more locations. Clearly, the same mechanismcannot be invoked in the absence of pressure influence on theshear response of a rubbery binder. Further investigationsneed to focus on the development of typical constitutive lawsof evolution and growth of hot spots in high explosives withthe ultimate aim of developing experimentally supported en-gineering models of detonation37–39 for use in large-scalecomputational codes.

ACKNOWLEDGMENTS

The authors would like to acknowledge the support ofthe U.S. Department of Energy through the ASCI-ASAP“Center for the Dynamic Response of Materials” ContractNo. B341492 under DOE Contract No. W-7407-ENG-48 atthe California Institute of Technology. The authors wouldalso like to thank Professor J. E. Shepherd at Caltech forrepeated helpful discussions during this work, to D. Nelsonand J. Campbell for preparing the final draft, and to S.Browne for help in preparing print-worthy figures.

APPENDIX

Expressions for nonzero terms ofAiab, Bi

ab, andRiab are

listed below. Further details can be found in Ref. 35. Forconvenience and clarity, the subscripti denoting individualrelaxation elements is omitted in all terms.

A11 =T11

l+

2me + le

l2 ,

A22 =le

l,

A33 =le

l,

B21 =T11

l+

me

l2 ,

B22 = T21 +k

lT11 +

k

l2me,

R11 =1

lsef fslS11T11 + S21S21 + meS11d,

R21 =1

2lsef fs4PS21T11 − 2T33S21 + 2meS21d,

R22 =1

lsef fflS22T22 + klS21T11 + lS21S21 + meslS22

+ kS21dg,

R33 =1

sef fS33sT33 + med.

1See, for example, C. H. Johansson and P. A. Persson,Detonics of HighExplosives(Academic, New York, 1970).

2By way of notation, the term “explosive” will be taken to refer to thecomposite explosive and the constituent phases will be referred to as “ex-plosive grain” and “binder.”

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9V. Boyle, R. Frey, and O. Blake, proceedings of the Ninth Symposium(International) on Detonation, 1989.

10G. M. Swallowe and J. E. Field, Proc. R. Soc. London, Ser. A379, 389(1982).

11M. E. Kipp, J. W. Nunziato, R. E. Setchell, and E. K. Walsh, proceedingsof the Seventh Symposium(International) on Detonation, 1981.

12S. N. Heavens and J. E. Field, Proc. R. Soc. London, Ser. A338, 77(1974).

13G. M. Swallowe and J. E. Field, proceedings of the Seventh Symposium(International) on Detonation, 1981.

14S. Y. Ho, proceedings of the Tenth Symposium(International) on Detona-tion, 1993.



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